12: Eigenvalues and Eigenvectors

Given only a vector space and no other structure, save for the zero vector, no vector is more important than any other. Once one also has a linear transformation the situation changes dramatically. Consider a vibrating string,

whose displacement at point \(x\) is given by a function \(y(x,t)\). The space of all displacement functions for the string can be modelled by a vector space \(V\). At this point, only the zero vector---the function \(y(x,t)=0\) drawn in grey---is the only special vector.